Anisotropic Hyperelasticity Formulations
Many materials of industrial and technological interest exhibit anisotropic elastic behavior due to the presence of preferred directions in their microstructure. Examples of such materials include common engineering materials (such as fiber-reinforced composites, reinforced rubber, and wood) as well as soft biological tissues (arterial walls, heart tissue, etc.). When these materials are subjected to small deformations (less than 2–5%), their mechanical behavior can generally be modeled adequately using conventional anisotropic linear elasticity ( see Defining Fully Anisotropic Elasticity). Under large deformations, however, these materials exhibit highly anisotropic and nonlinear elastic behavior due to rearrangements in the microstructure, such as reorientation of the fiber directions with deformation. The simulation of these nonlinear large-strain effects calls for more advanced constitutive models formulated within the framework of anisotropic hyperelasticity. Hyperelastic materials are described in terms of a “strain energy potential,” U , which defines the strain energy stored in the material per unit of reference volume (volume in the initial configuration) as a function of the deformation at that point in the material. Two distinct formulations are used for the representation of the strain energy potential of anisotropic hyperelastic materials: strain-based and invariant-based.
Strain-Based Formulation
In this case the strain energy function is expressed directly in terms of the components of a suitable strain tensor, such as the Green strain tensor (see Strain measures):
where εG=12(C-I) is Green's strain; C=FT⋅F is the right Cauchy-Green strain tensor; F is the deformation gradient; and I is the identity matrix. Without loss of generality, the strain energy function can be written in the form
where ˉεG=12(ˉC-I) is the modified Green strain tensor; ˉC=J-23C is the distortional part of the right Cauchy-Green strain; is the total volume change; and is the elastic volume ratio as defined below in Thermal Expansion.
The underlying assumption in models based on the strain-based formulation is that the preferred material directions are initially aligned with an orthogonal coordinate system in the reference (stress-free) configuration. These directions might become nonorthogonal only after deformation. Examples of this form of strain energy function include the generalized Fung-type form described below.
Invariant-Based Formulation
Using the continuum theory of fiber-reinforced composites (Spencer, 1984) the strain energy function can be expressed directly in terms of the invariants of the deformation tensor and fiber directions. For example, consider a composite material that consists of an isotropic hyperelastic matrix reinforced with families of fibers. The directions of the fibers in the reference configuration are characterized by a set of unit vectors , ( ). Assuming that the strain energy depends not only on deformation, but also on the fiber directions, the following form is postulated
The strain energy of the material must remain unchanged if both matrix and fibers in the reference configuration undergo a rigid body rotation. Then, following Spencer (1984), the strain energy can be expressed in terms of an irreducible set of scalar invariants that form the integrity basis of the tensor and the vectors :
where and are the first and second deviatoric strain invariants; is the elastic volume ratio (or third strain invariant); and are the pseudo-invariants of , ; and , defined as:
The terms are geometric constants (independent of deformation) equal to the cosine of the angle between the directions of any two families of fibers in the reference configuration:
Unlike for the case of the strain-based formulation, in the invariant-based formulation the fiber directions need not be orthogonal in the initial configuration. An example of an invariant-based energy function is the form proposed by Holzapfel, Gasser, and Ogden (2000) for arterial walls (see Holzapfel-Gasser-Ogden Form below).